rionyx * I went straight to the pawnshop and redeemed it with some money I had been going to spend on a birthday present for him.Q. Can I replace this bold sentence into 'some money I would have spent'?

Yes. I would add an adverb:'some money I otherwise would have spent'

rionyx * That large government programs are inherently bad is a shibboleth of the Republican party.Q. I don't know what the subject of 'is' is in this sentence.

That (subject) is a shibboleth of the Republican party.

Rewriting:The statement / The fact that large government programs are inherently bad is a shibboleth of the Republican party.

rionyx * Take pot luck: take a chance that whatever is available will be good or acceptable.Q. Is this sentence grammatically correct? I think 'take whatever chance (which is) available will be good or acceptable' is more appropriate.

Take a chance that whatever is available will be good or acceptable.

This sentence is fine. Yours is not, since the dependent clause is no longer intact.

Take (main verb, imperative mood) a chance (direct object) that (conjunction for the dependent clause) it (subject, dependent clause) will be (verb, dependent clause) good or acceptable.

rionyxsome money I had been going to spend ... 'some money I would have spent'?

The meaning is only slightly different.

Money you had been going to spend is money that you had planned or intended to spend.When you say "would have spent", you aren't talking about plans or intentions. In practical terms, however, the final result is the same. The money is gone!

rionyxThat large government programs are inherently bad is a shibboleth of the Republican party.Q. I don't know what the subject of 'is' is in this sentence.

The entire that-clause is the subject: That large government programs are inherently bad.

That large government programs are inherently bad | is | a shibboleth of the Republican party.

You can reverse the sentence and make a shibboleth of the Republican party the subject:

A shibboleth of the Republican party | is | that large goverment programs are inherently bad.

rionyxTake pot luck: take a chance that whatever is available will be good or acceptable.Q. Is this sentence grammatically correct?

It's not a sentence. It's a replacement paraphrase that serves as a definition. The implied sentence is this:

"To take pot luck" means "to take a chance that whatever is available will be good or acceptable".

rionyxI think 'take whatever chance (which is) available will be good or acceptable' is more appropriate.

No. You've changed from whatever (thing) is available to whatever chance is available. Don't change the original definition. It's fine.